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1: 28.17 Stability as x ±
§28.17 Stability as x ±
For real a and q ( 0 ) the stable regions are the open regions indicated in color in Figure 28.17.1. The boundary of each region comprises the characteristic curves a = a n ( q ) and a = b n ( q ) ; compare Figure 28.2.1. …
See accompanying text
Figure 28.17.1: Stability chart for eigenvalues of Mathieu’s equation (28.2.1). Magnify
2: 12.20 Approximations
Luke (1969b, pp. 25 and 35) gives Chebyshev-series expansions for the confluent hypergeometric functions U ( a , b , x ) and M ( a , b , x ) 13.2(i)) whose regions of validity include intervals with endpoints x = and x = 0 , respectively. …
3: 10.72 Mathematical Applications
In regions in which (10.72.1) has a simple turning point z 0 , that is, f ( z ) and g ( z ) are analytic (or with weaker conditions if z = x is a real variable) and z 0 is a simple zero of f ( z ) , asymptotic expansions of the solutions w for large u can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order 1 3 9.6(i)). These expansions are uniform with respect to z , including the turning point z 0 and its neighborhood, and the region of validity often includes cut neighborhoods (§1.10(vi)) of other singularities of the differential equation, especially irregular singularities. … In regions in which the function f ( z ) has a simple pole at z = z 0 and ( z z 0 ) 2 g ( z ) is analytic at z = z 0 (the case λ = 1 in §10.72(i)), asymptotic expansions of the solutions w of (10.72.1) for large u can be constructed in terms of Bessel functions and modified Bessel functions of order ± 1 + 4 ρ , where ρ is the limiting value of ( z z 0 ) 2 g ( z ) as z z 0 . These asymptotic expansions are uniform with respect to z , including cut neighborhoods of z 0 , and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation. …
4: 21.10 Methods of Computation
  • Belokolos et al. (1994, Chapter 5) and references therein. Here the Riemann surface is represented by the action of a Schottky group on a region of the complex plane. The same representation is used in Gianni et al. (1998).

  • 5: Bonita V. Saunders
    This work has resulted in several published papers presented as contributed or invited talks at universities and regional, national, and international conferences. …
    6: 5.10 Continued Fractions
    5.10.1 Ln Γ ( z ) + z ( z 1 2 ) ln z 1 2 ln ( 2 π ) = a 0 z + a 1 z + a 2 z + a 3 z + a 4 z + a 5 z + ,
    7: 14.31 Other Applications
    The conical functions 𝖯 1 2 + i τ m ( x ) appear in boundary-value problems for the Laplace equation in toroidal coordinates (§14.19(i)) for regions bounded by cones, by two intersecting spheres, or by one or two confocal hyperboloids of revolution (Kölbig (1981)). …
    8: 33.23 Methods of Computation
    Bardin et al. (1972) describes ten different methods for the calculation of F and G , valid in different regions of the ( η , ρ )-plane. … Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for F 0 and G 0 in the region inside the turning point: ρ < ρ tp ( η , ) .
    9: 19.7 Connection Formulas
    The first of the three relations maps each circular region onto itself and each hyperbolic region onto the other; in particular, it gives the Cauchy principal value of Π ( ϕ , α 2 , k ) when α 2 > csc 2 ϕ (see (19.6.5) for the complete case). … The second relation maps each hyperbolic region onto itself and each circular region onto the other: … The third relation (missing from the literature of Legendre’s integrals) maps each circular region onto the other and each hyperbolic region onto the other: …
    10: 19.21 Connection Formulas
    Change-of-parameter relations can be used to shift the parameter p of R J from either circular region to the other, or from either hyperbolic region to the other (§19.20(iii)). … For each value of p , permutation of x , y , z produces three values of q , one of which lies in the same region as p and two lie in the other region of the same type. In (19.21.12), if x is the largest (smallest) of x , y , and z , then p and q lie in the same region if it is circular (hyperbolic); otherwise p and q lie in different regions, both circular or both hyperbolic. …